criteria for validity of the maximum modulus principle for

Ark. Mat., 32 (1994), 121-155 (~) 1994 by Inst i tut Mittag-Leffier. All rights reserved

Criteria for validity of the maximum modulus principle for solutions of

linear parabolic systems

G e r s h o n I. K r e s i n a n d V l a d i m i r G. M a z ' y a ( 1 )

A b s t r a c t . We consider systems of partial differential equations of the first order in t and of order 2s in the x variables, which are uniformly parabolic in the sense of Petrovskii. We show tha t the classical maximum modulus principle is not valid in Rnx (0, T] for s>2 .

For second order systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle to hold in the layer R ~ x (0, T] and in the cylinder ~t x (0, T], where ~ is a bounded subdomain of R n. If the coefficients of the system do not depend on t, these conditions coincide. The necessary and sufficient condition in this case is tha t the principal part of the system is scalar and tha t the coefficients of the system satisfy a certain algebraic inequality. We show by an example tha t the scalar character of the principal part of the system everywhere in the domain is not necessary for validity of the classical maximum modulus principle when the coefficients depend both on x and t.

I n t r o d u c t i o n

I t is w e l l - k n o w n t h a t s o l u t i o n s of p a r a b o l i c s e c o n d o r d e r e q u a t i o n s w i t h r e a l

coef f i c ien t s in t h e c y l i n d e r

Q T = { ( x , t ) : x e ~ , O < t _ < T } , ~ c R ~,

s a t i s f y t h e m a x i m u m m o d u l u s p r i n c i p l e . N a m e l y , for a n y s o l u t i o n of t h e e q u a t i o n

Ou 0 u Ou - a ~ j ( x , t ) - x - - - x - - + E a ~ ( x , t ) - O - ~ x + a o ( x , t ) u = O ,

Ot i,j=l o x i o x j ~=1

(1) The research of the first author was supported by the Ministry of Absorption, State of Israel.

122 Gershon I. Kresin and Vladimir G. Maz'ya

where ((a~j)) is a positive-definite (n• function and a0>0, the inequality holds

lu(x,t)l _<sup{ lu(y, T)l : (y, T) e CgQT, 7 < T } .

This classical fact was extended to parabolic second order systems with scalar coefficients of the first and second derivatives in [10]. Maximum principles for weakly coupled parabolic systems are discussed in the books [8], [11].

Furthermore, there exists a large literature on "invariant sets" for non-linear parabolic systems (see, for example, [1], [2], [3], [4], [9], [12] and references there). However, we shall not characterize this interesting field, since here we consider only the maximum modulus principle and since the papers on invariant sets do not contain our results as special cases.

In this paper we find criteria for validity of the classical maximum modulus principle for solutions of the uniformly parabolic system in the sense of Petrovskii

Ou (1) ot F_, Am(x,t)o u=o

ImJ<2s

Here u is an m-component vector-valued function, A m are real or complex (m • m)- matrix-valued functions, 13= (~i,..., Dn) is a multiindex of order I/~I =/~i +... +/~, and

0 2- -0 Iml/Ox~ I ... Ox~". For s_> 1 the vector-function u is defined in the closure R~ +I of the layer R~+I=R~x (0, T]. In the special case s = l it will be defined also in the closure QT of the cylinder QT=~ • (0, T], where ~ is a bounded domain in R n.

Throughout the article we make the following assumptions: (A) The matrix-valued functions A m are defined in R~ +I and have bounded

derivatives in x up to the order ]DI which satisfy the uniform H51der condition on

R~ § with exponent a, 0<c~_< i, with respect to the parabolic distance

d [(x, t), (x', t')] = (Ix-x'12 + It-t'll/S) 1/2.

(B) For any point (x, t ) � 9 +~, the real parts of the )~-roots of the equation

det ( ( -1)~ E Am(x, t )am-AIm)--O Iml=2~

satisfy the inequality Re )~(x, t, or)~-51al 2~, where 5=const > 0 for any ~ � 9 n, I m is the identity matrix of order m, and I" I is the Euclidean length of a vector.

We obtain an expression for the best constant/C(R ~, T) in the inequality

lu(x, t)l <_/C(R ~, T) sup{ lu(y, 0)1: y �9 R n },

Maximum modulus principle for parabolic systems 123

where (x,t)cR~. +1. It is shown t h a t / C ( R = , T ) > I for all s_>2. For s = l , besides the constant IC(Rn,T), we study the best constant/C(~, T)

in the inequality

In(x, t)[ < ]C(~, T) sup{ In(Y, ~-)I: (Y, ~-) EFT },

where (x, t) E QT, FT = { (x, t) E OQT :t < T }. The closure F T of I~T is called the parabolic boundary of the domain ~ x (0, T).

Then we give separate necessary and sufficient conditions for validity of the classical maximum modulus principle (i.e. ]C(~,T)--1, )~ (R=,T)=I ) for solutions of the parabolic second order system

(2) Ou ou Ajk(x,t) + Aj(x,t) +A0(z,t)u=0. Ot ~ =

j , k= l

If the coefficients of the system (2) do not depend on t, then the above mentioned necessary and sufficient conditions coincide. More precisely, the following statement concerning the system

(3) 2 n Ou 0 u Ou

A j ~ ( 5 ) ~ + Z A t ( x ) ~ + Ao(x)u = o Ot

j , k = l 3 ~ j = l v x j

holds for the case of real coefficients.

T h e o r e m . The classical maximum modulus principle is valid for solutions of the system (3) in QT(R~ +1) if and only if:

(i) for all x e ~ ( x e R ~) the equalities

Ajk(x)=ajk(X)Im, l <_j,k<n,

hold, where ( (ajk ) ) is a positive-definite (n x n)-matrix-valued function; (ii) for all xE~ (xCR n) and for any ~j, ~eR m, j = l , . . . , n , with (~ j , ; )=0 , the

inequality

ajk (x)(~j, ~) + ~ ( A j (~)~, ~) + (~o (~)~, ~) > o j , k= l j = l

is valid.

The next assertion immediately follows from this theorem.


Coro l l a ry . The classical maximum modulus principle holds for solutions of the system (3) in QT(R~ +1) if and only if condition (i) of the theorem is satisfied and

(ii') for all xE~ (xER n) and any ~cR m, ]~1=1 the inequality

bij (x)[(.Ai (x)g, q)(.Aj (x)g, q) - (.A* (x)q, .A~ (x)q)] + 4(.A0 (x)q, q) ~_ 0 i , j : l

is valid. Here ((b~j)) is the (n• function inverse of ((aij)) and* means the passage to the transposed matrix.

In the paper we demonstrate by an example that the scalar character of the principal part of the system (2) everywhere in the domain is not necessary for validity of the maximum principle when the coefficients depend both on x and t.

Finally, it is shown that all the facts concerning the maximum modulus principle for solutions of systems with complex coefficients are corollaries of the corresponding assertions for systems with real coefficients.

In particular, for the scalar parabolic equation with complex coefficients

2 n Ou 0 u Ou ajk ( x ) ~ + ~ aj (x) z z - - +ao(x)u = 0

Ot j , k = l 3 k j = l uxj

we obtain in Subsection 2.2 that the classical maximum modulus principle is valid in QT(R~ +') i f and only if'.

(i) the (n x n)-matrix-valued function ((ajk) ) is real and positive-definite; (ii) for all xCD (xER n) the inequality

4 Re ao (x) > ~ bjk (x) Im aj (x) Im ak (x) j , k = l

holds. The present paper is related to our work [6] where we considered the system (2)

with constant matrix coefficients Ajk and with ,A0--,A1 . . . . . .An--0. In [6] we estab- lished that the classical maximum modulus principle holds if and only if.Ajk :ajkXm where ((ajk)) is a real positive-definite (n • n)-matrix.

We are going to devote a special paper to extend our present results to the so called maximum norm principle, where the role of the modulus is played by the norm in a finite-dimensional normed space.

Maximum modulus principle for parabolic sys tems 125

1. S y s t e m s o f o r d e r 2s in R T +1

1.1. T h e case o f real coefficients

1.1.1. Some notations. We introduce the operators

JgJ_<2s J~i=2s

where .A~ are real (m• functions satisfying the conditions (A) and (B), formulated in the Introduction.

Below we use the following notation.

Let S--Dx (0, Q], where D is a domain in R n and 0 < Q < e c . Let, for brevity,

either AJ = S or J~4 be the parabolic boundary of the domain D • (0, Q). By C(A4) we denote the space of continuous and bounded m-component vector-valued functions on AA with the norm

I]ul] = sup{ lu(q)l : q �9 AA }.

By C(k ' I ) (s ) we mean the space of m-component vector-valued functions u(x,t) on S whose derivatives with respect to x up to order k and first derivative with respect to t are continuous. By ck+a(R n) we denote the space of m-component vector-valued functions with continuous and bounded derivatives with respect to x

up to order k which satisfy the uniform H51der condition with exponent a. Finally,

let ck+'~'~/28(R~ +1) denote the space of m-component vector-valued functions with

derivatives up to order k with respect to x which are bounded in R~. +1 and satisfy

the uniform H51der condition on R~ +1 with exponent a with respect to the parabolic

distance. For the space of (m • m)-matr ix-valued functions, defined on R~. +1 and Ck+C~,a/2s ( Rn+ l ~ having similar properties, we use the notat ion m k T 1"

For s > 1 we put

(1.1) (Rn'T) = sup I]u]t=ollc(n.)'

where the supremum is taken over all functions in the class

C (2s,1) (R~+I)A C(R~. +1 )

satisfying the system OU Ot ra(x, t, O/Ox)u = o.

5 - Arkiv f'6r matematik


Let R~_ +1 = { (x, t) E Rn+l:t > 0 }. We introduce one more constant

Ilullc(R: ,l) (1.2) /C0(y) = sup iluit=olic(R.),

where the supremum is taken over all functions in the class

C(2S, 1) (R~+I)n C(R~ +1 )

satisfying the system ~u Ot P2o(y,O,O/Ox)u---O

and yER n plays the role of a parameter.

1.1.2. Representations for the constants ]C( R ~, T) and ]Co(y). According to [5] there exists one and only one function in the class

C(2S,1) [l:~n+l "~ n C(R~. +1 )

which is the solution of the Cauchy problem

Ou (1.3) Ot P.l(x, t, OlOx)u = 0 in R~. +1, u It=0 = r

with CEC(Rn). This solution can be represented in the form

u( x, t) =/n~ G( t, O, x, ~)r d~. (1.4)

Here G(t, T, x, ~) is the Green matrix (or the fundamental matrix of solutions of the Cauchy problem (1.3)). The Green matrix for the system

Ou Ot 9.1(y, t, O/Ox)u = 0

will be denoted by ~(t, 7-, x - ~ ; y). The Green matrix C0(t-T, x - ~ ; y) for the system

Ou Ot P.10(y,0,0/0x)u= 0

has the following representation

l~l=2s

M a x i m u m modulus principle for parabol ic sys tems 127

where a = (el, ..., an) E R n. This implies

Go(t--T,X--~;y) = (t-r)-n/2~P (t_-~/2~;y , (1.5)

with

P(x;y)=(27r)-n/R ei(X'~)exp[(-1)s E A~(y,O)aZJda. n 1~l=2s

When discussing the system (1) with coefficients depending only on t we use the notation

An(t), 92(t, O/Ox), ~o(t, O/Ox), G(t, % x-~), Co(t--T, x--~), P(x).

T h e o r e m 1.1. The following formula is valid

sup sup f (1.6) IE( Rn, T) = IG*(t,O,x,~?)zl d~, x E R n O < t < T Izl=l J R

where the * denotes passage to the transposed matrix. In particular,

(1.7) 1E0(y) = sup f IP*(,; y)z[ drl.

Let (x,t) be a fixed point in R~ +1. We find the norm ]u(x,t)] of the Proof. mapping C(R n) ~--+u(x, t)ER m, where u is defined by (1.4). Using the properties of the inner product in R "~ and the fact that the supremnm operations commute, we obtain

[u(x, t)l = s~p 1 /Rn G(t, 0, x, q)O(q) dr/

= s u p s u p (z, [ I~bl_<l Izl=l k, J R n

(1.8) P = sup sup ] (z, G(t, 0, x , , ) r dT?

]z]=l ]~b]~.l J R n

= sup sup f (G*(t,O,x, rl)z,r I z i = t l r J R ~

Let N(x,t)(z) denote the set of points ~?CR n on which

G*(t,O,x,v)z=O.

128 Gershon I. Kresin and Vladimir G. Maz 'ya

By (1.8) we have

sup sup f lu(x,t) l= (G*(t, O,x, ~)z, r ) d~. [zl=l Ir J R n \ N ( ~ , t ) ( z )

Clearly, the inside supremum of the last integral is attained at the vector-valued function

G* (t, 0, x, . )z

Consequently,

,u(x,t)[= sup ~ [G*(t,O,x,~)z[d~= sup s ,G*(t,O,x,~)z[d~. iz l=l n \ N ( ~ , t ) ( z ) iz l=l n

Using the definition (1.1) of the constant )U(R '~, T), we obtain

IC( Rn, T) ---- sup{ [[Ul[c(R~r): [[U[t=O[[C( R~) <_ 1,

Ou/Ot-P2(x, t, O/Ox)u = 0 in R~ +1}

= sup sup sup{ [u(x,t)[: [[Ult=O[[C(R~) <_ 1, x E R n O<t<_T

(1.9) Ou/Ot-Pg(x, t, O/Ox)u = 0 in R~. +1 }

= sup sup [u(x,t)[ x E R n O<t<_T

= sup sup s u p [ IG*(t,O,x,~/)z Id~ x E R ~ O<t<_T [z[=l J R n

which gives the representation (1.6). Substituting Go(t-T, x -7 ; Y) from (1.5) into (1.6) in place of G(t, O, x, ~), we

arrive at the representation of ~0(Y) in the form (1.7). []

1.1.3. Necessity of the condition s=l for the maximum modulus principle in R~ +1 . Henceforth all positive constants with non-significant value will be denoted by e with various indices.

The inequality

]~( R n, T) >_ sup{ K:o(y): y E R n }

L e m m a 1.1.

(1.10)

is valid.

Proof. From (1.6) it follows

(1.11) ]~(Rn,T)>lim sup sup s u p f IG*(t,O,y,~l)zld~h - - r---+O y E R n O < t < T [zt=l YBr(y)


where Br(y)={ x~R~:lx-yl<r }. We denote by IILII the norm

sup{ ILzl : z e R'% Izl = 1 }

r z-~a,c~/2s [ n n q - 1 \ of the (mxm)-mat r ix L. Since . A ~ m L~T }, then, according to estimates given in [5], the Green matrix G(t, T, y, n) for 7-=0 admits the representation

a(t, o, u, n) = Go (t, u- ,7; y) + [Go (t, y - n; n) - Go (t, y - n; y)] + w(t, y, n),

IIGo(t, y - n ; n)-Go(t , y - n ; y)ll [ (~ 28/(28-1) ] <_cxly-nl~t -'~/2~ exp -c2 ~

(1.12)

where

(1.13)

(1.14) F ( ly-nl sJ 2s 1

IlW(t, y, n)ll < c3 t-(n-a)/2s exp I-c4 - L \ t l - - ~ / j .

Using (1.12)-(1.14) and the representation (1.5), from (1.11) we obtain

/C(R ~, T) > sup sup yCR n I z l = l

= sup sup y C R ~ tz]=l

---- sup sup ycR~ lzl=l

= sup sup y E R n I z l = l

This and (1.7) yield the lemma. []

lira lim ~ f IG~(t,y-my)lzd~-c5r~-c6t ~ /~} r~O t~+o [.JB.(v)

lira lim JB t-'/2~ P ' ( ~ ; y ) z d n r---+O t-~+q~O r(Y)

lim lim f [P*(x;y)z[ dx r--~0 t ~ + O d B ~ t _ l / 2 ~

fro le*(x; y)zl ax.

inequality for /C(R~,T) given in the statement of the

L e m m a 1.2. If the classical maximum modulus principle is valid for solutions of the system

(1.15) Ou Ot E A~(t)O~u=O l_<l~l_<2s


in R~ +1, then A~(t)=a~(t)Im, where a~ are scalar functions.

Proof. 1. The structure of the matrix ~. From (1.8) and (1.9) it follows that

(1.16) 1 ]C( R'~, T) > (~*(t, O,x-~)z, r ) d~, sup sup I Izl=l Ir JR

where (x, t) is an arbitrary point of the layer R~. +1. Let z be a fixed unit vector in R m. Since

(1.17) /R~ G(t' O'x-rl) drl= Im'

(see [5]), then the set R n \ N(x,t)(z) has non-zero measure for any fixed z E R m, I z l= 1, and i x, t) ER~. +1 .

Suppose ]~ (R" ,T )= I and let there exist a unit m-dimensional vector z0 and the set MCRn\N(x,t)(zo), mesa M > 0 , such that for all ~ E M the inequality

z0 # P ( t , 0, x-~)zo/IP(t, o, x-~)zo I

holds. Then, using (1.16) and (1.17), we obtain

I = ] C ( R n , T ) > sup /R (G*(t,O,x-~?)Zo,r Ir

= sup I n (G*(t,O,x-V)Zo,r ]r J R \N(x, t ) (zo )

= /R (G*(t,O,x--~)Zo, G*(t,O,x--~)Zo) d~ ~\N(.,,)(zo) IP(t , 0, x-- r0zo I

fR * t > n\N(x,t)(zo)(~ (,O,x-~)zo,zo)dT?

= /R. (~*(t, O,x-~)Zo,zo) d~= 1.

Consequently, if ]C(R n, T)- - l , then for all zER m with Iz l=l and for almost all ~ERn\N(x,t)(z) one has

(1.18) z = P( t , 0, x-v)z/IG*(t, 0, x-v)zl.

Let gjk, j, k= l, ..., m, denote the elements of the matrix G. Setting zl-- (1, 0, ..., 0)*, ..., zm=(0, 0, ..., 1)* successively instead of z in (1.18) and taking into


account the continuity of the Green matrix G(t,T,X--r]) for t>T, we find that gjk(t, 0, x--r])----0 for j ~ k for all r]CRn\N(~,t)(zk).

Since gjk(t,O,x--r])=O for j = l , 2 , . . . , m and for r]EN(x,t)(zk), k = l , 2 , . . . , m , we conclude that gjk(t,O,x-r])=O for j ~ k and for all r]ER n. Now we put z'= m-1/2(1, 1, ..., 1)* instead of z in (1.18). Then

[g121 (t, 0, X--r])~-...-~-g2mm(t, O, X--r])] 1/2 = ml/2gjj(t, O, X--r])

for all j = l , 2 , . . . , m and for r]ERn\N(~,t)(z'). Hence making use of the equalities gjj(t, 0, x-r])-=O for j= l , 2, ..., m and for r]EN(x,t)(z'), we get

gl l (t, 0, x--~]) = g22 (t, 0, x - - r ] ) . . . . . gram(t, 0, x - - r ] )

for all r] C R n. Let g(t,O,x-~)--gjj(t,O,x-r]), l<_j<_m, and assume that ]C(Rn,T)=I. Then

the solution of the Cauchy problem for the system (1.15) has the form

(1.19) u(x, t) =/R~ g( t, O, x--r])r dr],

where r 0).

2. The structure of the operator 92. By r we denote a scalar function that is continuous and bounded on R n. Let

(1.20) uo(x, t) = / R n g(t, O, x--r])r (r]) dr].

According to (1.19) the vector-valued function hz(x,t)=uo(x,t)z, with zCR m, is a solution of the Cauchy problem (1.21)

Oh~ -92(t, c3/Ox)hz Oh~ Ot - Ot E A~(t)O~hz=O in R~. +1, hz It=0 =r l<l~l<2s

Setting z1=(1, 0, ..., 0)*, ..., z , ,=(0 , 0, ..., 1)* successively instead of z in (1.21), we obtain m 2 boundary value problems

(1.22) r OuO ~t~ p'q) R ~ q-1 , Ot E (t)O~uo = 0 in uo It=o = r 1_<1~1<2s

Here 5v, q is the Kronecker symbol, A (p'q) is the element of the matrix AZ placed at the intersection of the pth row and the qth column, p, q=l, 2, ..., m.


Suppose the initial function r in (1.20) has a compact support. Since the Green matrix g(t, % x-~])Im satisfies the inequality

F /ix_TiI2s\l/(2s--1)3 Ilg(t'T'X--rl)lmll=lg(t'T'x--rl)l<--cl(t--T)-n/2Sexp[--c2(' ~ ) ]

(see [5]), then (1.20) implies the estimate

lUo(X, t)] _< ca(t) exp(-c4(t)lx] :~/(2~-1)

for any fixed t > 0. Applying the Fourier transform with respect to the variables Xl, ..., xn to the

equation (1.22), we get

(1.23) 5p,qd(Fu~176 E il'iA(P'q)(t)a'=O" l_<iZl<2s

Let pCq. Since the function r determining u0 by (1.20) is arbitrary, the last equality yields A(P'q)(t)=O for all p~-q and all multiindices ~, 1_< [/31 <2s.

Suppose now that p--q. After integrating the equation (1.23) with account taken of the initial condition Fuo=Fr for t=0 , we find

Therefore,

Fr dT]

= Fr e x p ,

l_<l~l_<2s

A(P,P) (§ --h(q,q) {§ where p, q= l, 2, ..., m. Hence ~ ~oj -~Z ~oj for all p,q--1,2,...,m and for all multiindices /3, 1_<1~1_<2s. Thus, if E (R '~ ,T)=I , then the operator O/Ot- !~l(t, O/i)x) in the left-hand side of (1.15) satisfies the equality

Ou ~(t, O/Ox)u Ou ot = Z a (t)a u,

1_<$~]_<2s

where az are scalar functions. []


L e m m a 1.3. Let the parabolic system have the form

(1.24) Ou Ot E a~O~u=O, IZl=2s

where a/~ are constant scalar coefficients. Then the equality/(:(R ~, T ) = I is valid if and only if s= 1.

Proof. The sufficiency of the condition s--1 for the equality

~ ( R ~, T) = 1

follows from the positivity of the fundamental solution g0(t-T, x-r/) of the Cauchy problem for the parabolic second order equation with constant real coefficients

Ou ~ ajk 02u

Ot Oxj Oxk j , k : l

m - 0

and from the formula

(1.25) u(x, t) = f go(t , x - r / ) r dr/ d R n

for the solution of the Cauchy problem

Ou f i __02u _ R~+I ' Ot ajk OxjOxk 0 in u(x, O) = •(x),

j , k = l

where u and r are m-component vector-vMued functions, r E C(Rn). Necessity. The solution of the Cauchy problem for the system (1.24) can be

expressed by (1.25) for any s, where

is the fundamental solution of the Cauchy problem for the scalar equation (1.24), i.e. for m--1. Consequently, the constant

19o(t,x-r/)ld = fRo IP( )ld does not depend on m and one can put m = l .

134 Ger shon I. Kres in a n d Vladimir G. M a z ' y a

We note that the solution of the Cauchy problem

(1.26) Ov 02~v Ot a1""10--~ ----0 in R~, V(Xl,O)=r

where sign el . . .1--(-1) s-1 and v=v(xl,t) is a scalar function, also satisfies the Cauchy problem for the scalar equation (1.24) with the initial condition u(x, 0)= r Therefore,

~(R", T) _> k = snp(llvllc(~)/llv I+=o IIc(R'>),

where the supremum is taken over all solutions of (1.26) in the class

C(2S,1)(R~)nC(~).

Henceforth in this lemma xl is denoted by x. We set

a = {al...ll 1/2~.

The solution of the Cauchy problem

C~2S V Ov 4_(_1)~a2~ = 0 in R 2, v(x,O)=r Ot Ox 2~

r has the form

f c ~ x - r ] V(X,t) =t-1/2s ~-oo P (tl--7~) r dr1' (1.27)

where

i F e ~x~ exp(-a2Slal2S ) d a P(x) = ~ oo

/? _ 1 c o s ( x ~ / e x p ( - a ~ " ~ s / d ~ --71"

__ 1 / ? oxp(_o

From (1.27) it follows that


Since

t -~128 P drl = P(rl) dr I = 1, , 1 - o o o o

then k > l in case P(rl) changes sign. We show that P(rl) is a function with alternating signs. Put

(1.28) U~(r) = cos(rO) exp(-e ~) dO,

where r>0, A_>I. Then P(x)=(rca)-l.F28(x/a). Since ~ ( 0 ) > 0 , then 5ra(r)>0 with re(0,s] for some s>0. We verify that 9v~(3rr) <0 for A>4.

Integrating by parts in (1.28), we find

~0 (x) 1 exp(-e ~) d(sin(re)) f~(r) = 7

/o = i exp(_e~) sin(re) ~ +-~ o ~-* e x p ( - e ~) sin(re) de r o r

-- -~ e ~-1 e x p ( - e ~) sin(re) dO = (&(r )+L~(r ) ) , r

where

Ja (r ) = el2 e ~-1 exp(-e~) sin(re) de,

L~(r) = 0 ~-1 exp(-O ~) sin(rO) dO.

The maximum value of the function f ( e ) = e ;~-~ exp(-e ~) is attained at Co= ((A- 1)/A) 1/~ < 1. Hence na (311") <0 for )~> 1, so it suffices to show that Jx (3rr) <0 when A > 4.

Estimating each of four integrals in the equality

=/'~"~ e '-~ exp(-e ~') sin(~,,e) de

1 + f2/3 eX-1 exp(--e'x)sin(afro) de+ f l 4/a e "x-1 exp(-e ~) sin (at re) de

/42 + o ~-1 exp(-O~)sin(~O) CO, /a

136 Ger shon I. Kres in and Vlad imi r G. M a z ' y a

we get 1

J,x(aTr) <_ f2/ao'x-lexp(-O'X)dO+(w ~) f sin(37r0)dO JO J 2 / 3

+ (4)~-1 exp(-(34-) "x) [4/3sin(37rO ) dO+f44 2 0 A-1 exp(-O~)dO J1 /a

= ~-l(1-exp(-(F))- s (F- lexp( - (F) (4)a-1 exp (_ (4)a)+X-1 (exp (_ (~)x) _ exp(_2x)/

_< (2)a (A-l-~r-1 exp(- (2)a))

A-1 2 +( - 5-~ (4)~-1) exp ( - (4)~) -/~-1 exp(-2;~).

The functions

f1()~)=~-1-Tr-1 exp(-(2) ~), f2(~) = ~ - 1 - ~ 2 (4~X-13r

are monotone decreasing as A increases. Since f1(4)<0, f2(4)<0, then J~(37r)<0 for A_>4. Taking into account that

J:~(r) = (A/r)(J~(r)+L~(r)), P(x) = (Tra)-lJc2s(x/a), L~(37r) < 0 for A_> 1, and .T'x(0) > 0,

we arrive at the conclusion that P changes sign for s > 2. Thus, ~(R n, T)> k> 1 for s>2. []

Theorem 1.2. The classical maximum modulus principle is not valid for solutions of the system

Ou ~(x, t, O/Ox)u = o in R V 1

Ot / f s> l .

Proof. Lemma 1.2 implies that the equality/(:o(y)=l is valid for the system

Ou ot ~ ~(y,O)O~u=o

l~l=2s

only if ~4~(y, 0)=a~(y, O)Im. By Lemma 1.3 we have/Co(y)>l for the system

Ou ot ~ a~(y, 0)0~=0,

1~1=28

with s> 1 which together with (1.10) completes the proof of the theorem. []


1.2. T h e case o f c o m p l e x coeff ic ients

In this subsection we extend basic results of Subsection 1.1 to the systems (1) with complex coefficients and with solutions u=v+iw, where v and w are m- component vector-valued functions with real-valued components.

For the spaces of vector-valued functions with complex components we retain the same notation as in the case of real components but use bold. The same relates notation for the spaces of matrix-valued functions.

We introduce the operators

~(x,t,O/Ox)= ~ X~(x,t)o~, ;~o(x,t,O/Ox)= ~ A~(x,t)o~, 1/31_<2s I/~l=2s

where ,4~ are ( m x m)-matrix-valued functions with complex elements satisfying the conditions (A) and (B), formulated in Introduction.

Let ~ and ?-/Z be real (m • m)-matrix-valued functions defined on R~ +1 such that .An(x , t)=TC~(x, t)+iT-l~(x, t). We use the following notation

~(x,t, o/o~) = Z 7e,(x,t)o~, 1~l_<2s

~to(x,t,O/Ox)= ~ 7e~(x,t)og, p/31=2~

~(~, t, a/ax) = ~ ~,(~,t)a~, IZI<2~

SSo(~,t,a/a~)= ~ ~(x,t)o~.

Separating the real and imaginary parts of the system

Ou Ot ~ ( x , t, O/Ox)u = 0,

we get the system with real coefficients, which like the original system is uniformly

parabolic in R~ +1

Ov ot ~(x, t, o /Ox)v+~(x, t, O/Ox)~ = o,

Ow Ot 2)(x,t,O/Ox)v-~(x,t,O/Ox)w=O.

Remark 1. The preservation of the uniform parabolicity in the sense of Petro- vskii under the passage from a system with complex coefficients to the system with real coefficients is a corollary of the following simple algebraic property.

Let Q be an (m • m)-matrix with R and H being its real and imaginary parts, respectively. Let the eigenvalues A of Q satisfy R e A < - 6 , 6>0. Then for the eigenvalues # of the matrix


the inequality R e # < - 5 holds. In fact,

det(A-#I2m) = det ( R - # I m \ H

= det ~ Q - #Im \ H

R - #Ira = det H R - #Im

0 = d e t ( Q - ~Im) d e t ( Q - #Ira),

/

where Q is a matrix whose eleme ts are complex conjugate of corresponding elements of Q. Since

det( Q - #Im ) = det(Q-f~Im) = det(Q-f~Im),

then Re # < -5 . We introduce the matrix differential operators

~(x, t, o/o~) = ( ~(x, t, o/o~) \ ~)(x, t, O/Ox)

~(x , t, OlOx) = ( ~o(x, t, OlOx) \ ~o(x, t, OlO~)

-~(x, t, alax) ~(x, t, alaz) ) ' -~)0 (x, t, o/ox) ~o(~, t, a/ax) )

Let G'(t, % x, 7) and G~( t--T, X--q; y) denote fundamental matrices of solutions of the Cauchy problem for the systems

0 0 0 ( ~ - ~ ( x , t , O / O x ) ) { v , w } = O and (~--~-~o(Y, ,O/Ox)){v,w}=O,

respectively. Further, let pt(x, y) be the (2mx2m)-matrix-valued function in the representation

t 2s;Y "

Let J~4 be the set in the Euclidean space introduced in Subsection 1.1. The norm in the space C(J~4) of continuous and bounded on J~ vector-valued functions u=v+iw with m complex components is defined by the equality

Ilul[ = sup{ (Iv(q)12 + ]w(q)]2)U2 : q e M }.

As in the definition of )U(R ~, T) we put

](:'(R '~, T) -- sup IlUtlc(R~"+I) Iru I~=0 llc(Ro) '


where the supremum is taken over all functions u = v +iw in the class C(2S'1)(R~+I)N

C(R~. +1) satisfying the system

Ou Ot

We define one more constant

---s t, O/Ox)u = o.

ll~llc(.~+l) /C~ (y) ---- sup ii u It=0 IIC(nn) '

where the supremum is taken over all solutions u = v + i w of the system

Ou Ot 2.o(y,O,O/Ox)u=O

in the class C(2s'I)(R~+I)nC(R~ +1) and yER n plays the role of a parameter.

Clearly, the constant ~'(Rn, T) for the system

Ou Ot 2.(x, t, O/Ox)u = 0

coincides with the constant/C(R ~, T) for the system

Therefore, all the assertions concerning K)(R~,T) are immediate corollaries of analogous assertions about ~(Rn ,T) . Taking this into account, we obtain the assertions marked below by primes from Theorems 1.1, 1.2 and Lemma 1.1.

T h e o r e m 1.1 ~. The following formula is valid

IC'(Rn,T)= sup sup sup /" I(G')*(t,O,x,~)z[d~, x c R n O<t<_T {zER2m:[zI=l } J R ~

where the * means passage to the transposed matrix. In particular,

t:~(y) = sup {zcR2"~:lzl=l}

L e m m a 1.1' . The inequality

holds.

T h e o r e m 1.2' .

lutions of the system

if s> l .

/R~ ](P')*(~; Y)Zl d~l.

/C'(R n, T) _> sup{/C~(y) : y �9 R ~ }

The classical maximum modulus principle is not valid for so-

Ou ~(x, t, O/Ox)u = 0 in R~. +1 Ot

140 Gershon I. Kresin and Vlad imi r G. M a z ' y a

2. Second o rde r systems

2.1. T h e case o f real coefficients

2.1.1. Necessary conditions. In this subsection we study the validity of the classical maximum modulus principle for the system (2) with coefficients ~4jk, Aj,

.,40 (1 < j , k < n) that are real (m x m)-matrix-valued functions in the layer R:~ +1 and in the cylinder ~)T=~X [0, T], where f~ is a bounded domain in R n.

We retain all notations introduced in Section 1 for arbitrary s > 1 and introduce one more constant

(2.1) ~ ( ~ , T ) = sup Ilullc(c~T) ]lu let IIc(~T)'

where the supremum is taken over all solutions of the system (2) in the class C(2'I)(QT)nC(~2T).

L e m m a 2.1. The inequality

(2.2) 1C(f~, T) _> sup{/C0(y): y �9 f~ }

holds.

Proof. Let y be an arbitrary point of ~ and let the radius of the ball Br(y) be so small that Br(y)C~. Further, let

r ]r suppr 0<r

The vector-valued function

(2.3)

is the solution of the Cauchy problem

2 n . 0 u~ v-~ Ou~ Ou~o~_ j,k=l A j k ( x , t ) ~ + ~= Aj(x,t)~xj +Ao(x,t)u~ = 0

u~(x,O)=r

in R~, +1,

Since r ~4j,.Ao e C~'a/2(R~ +1), then, according to [5], one has

(2.4)

Maximum modulus principle for parabolic systems

The last estimate used for (x, t) C (~\B~(y)) x (0, T], yields

_ exp~-c2~-[---)dr]

~ c 3 c n t - n / 2 e x p ( - - C 4 ~ )

which implies

141

lug(x, t)l < c ~ n,

where lu~(x,t)[ is the norm of the mapping r Hence, the following estimate is valid for sufficiently small

(2.5) sup{ [ue(x, t)[: (x, t) e 0 a x [0, T] } _< 1.

From this and the definition (2.1) of the constant ~ (~ , T) we get

(2.6) ]~(~t, T) > sup lim lim [u~(y, t)[. - - y E ~ e - - -*0t - - -*+0

Using (2.3) and (2.4), in the same way as in the proof of Theorem 1.1, we find

lu~(y,t)l-- sup f IG*(t,O,y,~)zld~. Izl=l JB~(y)

The last equality and (2.6) yield

~ (~ , T) _> sup lim lira sup f IG*(t,O,y,~)zld~? yE~ e-~O t--~+O izl= 1 J B~(y)

(2.7) _>sup sup l~ ~m [ la*(t,o,y,~)zl~

It was shown in Lemma 1.1 that

lim lim f IG*(t,O,y,~)zld~ s u p Izl=X ~ot~+o JB~(y)

= sup lim chm f IG~(t,y-~;y)zld~=lCo(y ) ]zl=l ~ Ot +OJB~(y )

which together with (2.7) gives the lower estimate for K:(~, T) in the statement of the lemma. []

142 Ge r shon I. Kres in and Vlad imi r G. M a z ' y a

T h e o r e m 2.1. The classical maximum modulus principle is valid for solutions of the system

(2.8) Ou uxjuxkO2u ~ ) ~--~-~ Ajk(t) Aj(t = 0 j , k = l "= J

in R~ +1 if and only if the equalities

Ajk(t)=ajk(t)I,~, Aj(t)=aj(t)Im, l < j , k < n ,

hold, where ((aj~)) is a positive-definite (nxn)-matrix-valued /unction and ay are scalar/unctions.

Pro@ The necessity of the above equalities follows from Lemma 1.2. We show that K(R n, T)--1 under the conditions of the theorem.

Consider the Cauchy problem

0 u Ou R~+I ' (2.9) Ou ajk(t) + aj(t) = 0 in 0t

j , k = l

uIt=o=r where r The solution has the form

u(x, t) = f , . g(t, 0, x-•)r &?,

where g(t, ~-, x -~) is the fundamental solution of the Cauchy problem for the equation (2.9) in which u is a scalar function.

Substituting g(t, O, x-~)Im in place of G(t, O, x -~) in (1.6), we obtain

sup sup / ,g(t ,O,x-,)]d,= sup / ~:(Rn,T) = Ig(t,0, )l xE R n O<t<_T J R n O<t<_T J R n

which means that ~(Rn,T) does not depend on m. Since ]C(Rn,T)=I for m = l we arrive at the sufficiency of conditions of the theorem. []

T h e o r e m 2.2. Let the classical maximum modulus principle be valid for solutions of the system (2) in QT(p~+I). Then:

(i) for all xE• (xER n) the equalities

Ajk(X,O)=ajk(x)Im, l < j , k < n ,

hold, where ( (ajk ) ) is a positive-definite (n x n)-matrix-valued /unction; (ii) for all x e ~ (xeR ~) and/or all ~, ~eR m, j = l , . . . , n , with (~j,~)=0 the

inequality n

_> 0 j,k=l j=l

is valid.


Proof. We assume for brevity that ~ is a bounded subdomain of R '~ or f~= R ~. From Lemmas 1.1, 2.1 it follows that the equality K:(f~,T)=I is valid only if K:0(y)=l for all yESt, where/Co(y) is the constant (1.2) defined for the system

�9 02u o~at ~ AJk(Y'O) ox----gx~

j ,k=l

By Theorem 2.1 the equality/Eo(y)--1 takes place if and only if (i) is satisfied. Now we prove the necessity of (ii). Let y be an arbitrary fixed point of f~,

and let radius of the ball B~(y) be so small that B,(y)Cf~. We introduce the vector-valued function

v(x)= jxj+ ) _ +1 12) ,

where ~jeR m, qeRm\{0}, (~j, ~)--0, j - - l , ...,n. F~rther, let

xeeC~176 X~(x)=l for Ixl<_~/2, X~(x)=0 for Ixl_>~

and O<_x~(x)_<l for a l l x E R n,

where ~ < r. The vector-valued function

(2.10) u~(x, t) = f G(t, O, x, ~)X~(~I-y)v(~-y) dTh J R n

where G(t, r, x, 7) is the Green matrix of the system (2), is the solution of the Cauchy problem for the system (2) with the initial data u~(x,O)=v~(x), where ve(x)=x6(x-y)v(x-y) . If ~2 is a bounded subdomain of R n, then by (2.5) the value of e can be chosen so small that

sup( [u~(x, t)]: (x, t) e 0~ x [0, T] } _< 1.

Then [[u6 [Pr [[c(~r) = 1, and, consequently

(2.11) /(:(f~,T)> sup [ue(y,t)l. O<t<_T

In the case f l=R n the last inequality is obvious.


Suppose the condition (i) is satisfied. We take the scalar product of the system (2) with u and transform the equality

ou) - - , u - a~k(x) ,u ,u

~ Ot ) j~k~ l ~ OXjOXk j = l

) Ou + ( A o ( x , t ) u , u ) - ~ ( [A jk (x , t ) - a jk (x ) I~ ] ~m~j?~ ,u =0,

j,k=l \ ~JUZCk

where ajk(x)Im=Ajk(X, 0). As a result we find

l Olu, 2 _ l ~ ayk(x) O2lul2 ~-~ ajk(x)( O~xj OU ) 2 0 t 2 OX~OXk ' OXk j,k=l j,k=l

j = l

+ Z ([~j~(x, n t ) - a j k (x)Im] ~ , u ) . 02u j,k=l

Since the coefficients of the system (2) belong to the class C~'~/2(R~ +~) and since

v~ EC2+~(R ~) then, according to [5], u~ EC2+~'~/2(R~+~). So after substitution of u~ defined by (2.10) into (2.12) in place of u, we obtain

lim c3lu~12 - 2{ 2 (~,t)--.(y,+o) Ot

o~lv~l ~ ( ov~ ov~ ~ ajk(Y)Oxj-----~xk ~ ajk(Y)\OXj OXk]

j , k = l j,k=l

--j~=l (Aj(y, " OVe ' O)V~,Ve)} n o ) ~ , v~) -(re(y, ~ :

Hence, using the equalities

O~lv~l~ x = 0 , OXgXk =y

where j, k = l , ...,n, we find

Ov~ = ~5, v~(y)= ~, ~Xj x=y

(2.13)

lim t--~+o Ot o l u ~ ( y , t ) r 2 _

1~12 ajk(y)(~j,~k)+ Aj(y, 0)~j,~) L j,k=l j = l

+ (Ao (y, 0)~, ~)1'


The function lug(y, t)l 2 is continuous on [0, T] and differentiable at all points of the interval (0, T), its derivative Olu~(y , t)12/Ot tends to a finite limit as t--~+0. Therefore, the function l u~ (y, t)]2 has the right-hand derivative cO+ l u~ (y, t)12~cOt at t=O, and by virtue of (2.13) we have

(2.14) ~ 2 ajk(y)(~j, ~k)+ (.Aj (y, 0)~j, ~)

1 12 5 , - 1 5:1

+ (A0(y, J

Suppose )U(~,T)--1 and let there exist a point y E ~ and v e c t o r s ~jcR m, ; e R m \ { 0 } with (~j , ; )=0, j = l , ...,n, for which

(2.15) n

ajk(y)(~j, ~k)+E(.Aj(y, O)~j, ; ) + (Jio(Y, 0)q, q) < 0. j,k=l j = l

Since lu~(y,O)l=lv(O)l=l, then (2.14) and (2.15)imply the existence of a constant 5>0 such that lu~(y,t)l>l for 0 < t < 5 . From this and (2.11) it follows that /C(~, T) > 1 which contradicts our assumption on the validity of the classical maximum modulus principle.

Thus, i f / ( : (~ ,T )= I , then for all x e ~ and for all vectors ~jCR m, qCRm\{0} with (~j, ~)=0, j = l , ..., n, the inequality

n

ajk(x)(~j, ~k)+E(~4j(x, O)~j, r (~40(x, 0)r ~) _> 0 j,k=l j = l

holds. The condition ~ER "~ \{0} can be omitted since the inequality

ajk(x)(r Ck) > 0 j,k=l

is true due to the necessity of the condition (i) for validity of the classical maximum modulus principle. []

Remark 2. In what follows we show that conditions (i), (ii) of Theorem 2.2, are necessary and sufficient for validity of the classical maximum modulus principle for second order systems with coefficients depending only on x. But in general, when the coefficients depend on x and t these conditions are not sufficient.


Consider, for example, the parabolic system

(2.16) Ou ~-~ . . . 02u ~ t) Ou Ot A J k ( x ' t ) ~ + E Aj(x' = 0

j , k= l j = l OXj

in R~. +1, where Ajk -'~2+a'a/2~nn+l~ A _.~l+~,~/2~nn+h ~tJm ~1~ T ), r ~s T ). Suppose the coefficients of the system (2.16) do not depend on x in the layer R~ +1, 0 < 5 < T , and let

Ayk(X,O)=ajkXm, l j(x,O)=ajIm,

where ((ajk)) is a positive-definite (n x n)-matrix and aj are scalars. Let the matrix A1 (x, t) be non-diagonal for all (x, t ) 6R~ +1. Then, according to Theorem 2.1 the classical maximum modulus principle is not valid for the system (2.16) in R~ +1 (the more so in R:~ +1) whereas the conditions (i), (ii) of Theorem 2.2 are satisfied. []

2.1.2. Sufficient conditions for systems with scalar principal part. Next we present a theorem on a sufficient condition for validity of the classical maximum modulus principle for second order systems with scalar p r inc ip i part.

T h e o r e m 2.3. Let the coe~cients of the parabolic system

(2.17) O u ~ 02u ~ A j )~--~j Ot ajk (x, t ) ~ + (x, t +.Ao(x, t)u = 0 j,k=l j = l

satisfy the condition: (j) for all (x, t)eQT((X, t)eR~. +1) and for all vectors ~j, ~eR m with (~j, ~)=0,

j = l , ..., n, the inequality

(2.18) n

ajk(x, t)(~j, ~k )+E(Aj (x , t)~j, ; )+(A0(x , t);, ~) >_ 0 j,k=l j=l

holds. Then ]C(~t,T)=I ( ]C(R~,T)=I) .

Proof. Suppose first that for all (x,t)EQT and for all ( j eR "~, ~eRm\{0}, j = l , ..., n, with (~j, ~)=0 we have

(2.19) n

ajk(X, t)(~j, ~k)+E(,4j(X, t)~j, ; ) + (Jio(X, t);, ;) > O. j,k=l j = l

We show that for any non-trivial s o l u t i o n uEC(2,1)(QT)NC(QT) of the system (2.17) the function lu(x,t)] can not attain its global maximum at a point


(x,t)EQT. This will imply that if the system (2.17) has a regular solution in QT and if (2.19) holds, then the function in(x, t)] takes its maximum value on FT.

From (2.17) we have

(2.20)

(< ) 10tuB 2 1 021ul 2 _Eajk(X, t ) OU OU 2 0 t -- 2 ajk(x't) OxjOxk j = l ' OXk

j , k=l

-Z j = l \

Suppose the function ]u(x,t)l takes its global maximum at a point (xo,to)EQT. Then

(2.21) O[u]2 = 2 ( O~--~j ) ~u =0~ Oxj (xo,to) (xo,to)

(2.22) ~ : 2 ( ~ ) (xo,to)- Ot (~o,to) \ ot ' U >0,

02Jul2 (zo,to)<_O. (2.23) ~ ajk(Xo, t o ) ~ j , k= l

By (2.20)-(2.23) we have

j,k=l ajk(x~176 Oxj' ~-Xk

Ou

which contradicts (2.19) and hence In(x, t)l can not attain its global maximum at (Xo, to) E QT. This implies

Suppose now that (2.18) holds under the conditions in the statement of the theorem. Let e- -const>0 and let u be the solution of the system (2.17) in the class C(2'I)(QT)nC(QT). The vector-valued function v(x,t)=u(x,t) exp( -e t ) is the solution of the system

n ov ajk(x,t) Ov

j , k = l j = l

=O.


According to our assumption, for all (x,t)EQT and all ~jCR m, ~cRm\{0} with (~j, g)=0, l< j<n , we have

ajk(x,t)(~j,~k)+E(Aj(x,t)~j,~)+(Ao(x,t)~,~)+~l~l > 0. j,k=l j = l

By what we proved above

and hence

Consequently,

max le-~tu(x, t)I = max le-~tu(x, t)I. Qr ~T

IMIc(0T) < e Tll IIcr

Since a is arbitrary, the best constant in the last inequality is equal to one. Now we turn to the constant K(R n, T). Contrary to the case of the bounded

domain, one can not immediately conclude that ]C(R '~, T ) = 1 because of the absence of the global maximum of the function I-ix, t)l in +1

Suppose the inequality (2.18) holds for all (x, t) 6 R~. +1 and all ~j, q 6 R m with (~j, q)=0, j = 1, ..., n, and that the classical maximum modulus principle is not valid for solutions of the system (2.17) in R~. +1. Then there exists a point (x0, to)6R~- +1 and a vector-valued function C E C(R"), Ir 1 such that

(2.24) lu(xo, to)l = f G(to, O, Xo, ~)r d~ > 1. JR n

By (2.4) one can assume that r has a compact support, supp r If ]X-XoI > R > p then (2.4) implies

/R n a(t,O,x, Tl)~)(~)d~'] ~_Cl e x p ( - c 2 (R-@)2 ), where O<t<_T, xER n \Bn(xo).

Applying the assertion of the present theorem for a cylinder with a bounded base, we get

lu(x't)l = ]R- a(t'O'x'v)r <- 1,

where (x,t)EBR(Xo)x [0, T] and R is sufficiently large. The last inequality contradicts (2.24) which proves the validity of the classical maximum modulus principle in R~. +1. []

2.1.3. Necessary and sufficient conditions. Theorems 2.2 and 2.3 immediately imply the following assertion.


T h e o r e m 2.4. The classical maximum modulus principle is valid for solutions of the system (3) in QT(R~ +1) if and only if:

(i) for all x e ~ ( x e R n) the equalities

Ajk(x)=ajk(x)Im, l<_j ,k<m,

hold, where ( (ajk ) ) is a positive-definite (n x n)-matrix-valued function; (ii) for all x e ~ ( x e R n) and all ~j, r m, j = l , . . . , n , with (~ j , ; )=0, the

inequality

n

(2.25) ~ ajk(xl(~j, ~kl+E(Aj(x)~j, ~)+(A0(x)~, ~1 > 0 j , k=l j = l

~s valid.

Theorem 2.4 implies

C o r o l l a r y 2.1. The classical maximum modulus principle is valid for solutions of the system

~ ~ A j ( x l_~x ~ Ou 0 u Ou Ajk(x)~--Z:~ + ~ =0 Ot v~Jt]~k j = l j,k=l

in QT(R~ +1) if and only if for all xei2 (xCR '~) the equalities

hold, where ( (ajk ) ) is a positive-definite (n x n)-matrix-valued function and aj are scalar functions.

Proof. Putt ing .40=0 in (2.25) we get

n

j ,k=l j = l

which can be valid for all xEt2 (xCR n) and for all ~j ,r "~ with (~j , ; )=0, j = 1, ..., n, only provided (Aj(x)~j, r Consequently, ~4j(x)=aj(X)Im, where

aj(x)=jl~l'l)(xl-- A~2'2)(x) . . . . . Jl~m'm)(x), j = l,...,n. []

Remark 3. Minimizing the left-hand side of (2.25) over

~=(~l , . . . ,~n) , ~jER "~,


for a fixed gERm with (~j,~)=0, j--1, ..., n, one can write the condition (ii) of Theorem 2.4 in another form. This was used in [7], where the maximum modulus principle was studied for elliptic systems with scalar principal part.

One may assume that ;ERm\{0} , since the inequality

• ajk(x)(~y, ~k) >_ 0 j , k = l

is provided by the condition (i) of Theorem 2.4 for all f j ER m. Let

n �9 )Fq (~1, "", i n ) = a j k ( x ) ( ~ j , f k ) + E ( . A j ( x ) f j, g ) + (.A0(x)g, ~)

j , k = l j = l

or, which is the same

�9 ~ ( f l , i n ) (i) (i) "'" = E ajk(x)fj ~k + E Aj(i'k) (~,,j~,c(k)~(i), j , k=l i=1 j = l i ,k=l

+

i ,k=l

where ~k) and g(k) are components of the vectors f j and ~, respectively. At a point of the constraint extremum of the function 5r~(f~, ..., in) one has

(2.26)

of~i ) "~"q ( f l , "-', in ) - - E /~Jf} ~) ~-(z) j = l i=1

a / x ~e(i) -~ ~ ..4 (j'i) { x ~ (j) j k k }qj ~ k ~ ] -- k q ( i ) = v , ~ N ~ 2 /=1 j=l

where k = l , 2, ..., n, i=1 , ..., m and the following constraint relations are valid

m EfJi)4 ) =0, j = 1 , 2 , ..,n. i=1

Multiplying (2.26) by ~(i) and summing up over i from 1 to m we obtain


where * means passage to the transposed matrix. Consequently, the conditions (2.26) defining ~j can be written in the form

n m

2 E ajk(x)~J i)+ E g, j = l j = l

Taking into account the symmetry of the matrix ((ajk(x))), we find

( 2 . 2 7 ) 1 ~ bjk(x)[igl_2(A~(x)g ' g)r k=l

where j = l , 2, ... n and ((bjk(x))) is the inverse matrix of ((ajk(X))). The function 9vr (~1,-.., ~n) attains its constraint minimum at the vectors (2.27)

because of the positive-definiteness of the matrix ((ajk(x))). Calculating the value of 9rr ..., ~,) at vectors (2.27), we obtain

min{ 9rr (~1, ..., ~ ) : ~1, ..., ~ e R m, (~1, g) = 0, ..., ( ~ , q) = 0 }

= 11 1-2 g) i , j= l

1 4 E bij(x)(A*(x)g, A~(x)g)+(Ao(x)g, g).

i , j= l

Thus Theorem 2.4 implies the following assertion.

C o r o l l a r y 2.2. The classical maximum modulus principle is valid for solutions of the system (3) in QT(R~ +1) if and only if the condition (i) of Theorem 2.4 is satisfied and

(ii') for all xef~ ( xeR n) and for any geR m, H = I the inequality

> 0 i , j= l

holds, where ( ( bij ) ) is the (n x n )-matrix-valued function inverse of ( ( aij ) ) and .A ; (x) is the matrix transposed of Aj(x).

Remark 4. We give an example of a system whose principal part is not a scalar differential operator in the whole domain and for which the classical maximum modulus principle is valid in R~ +1.


Consider the parabolic system

(2.2+ oV+Ao(x, )v+;v=O Ot AJk(x't) OxyOx~+ Oxj j , k : l

A ~r~2H-c~,c~/2zr)nH-l~ J ~fy1q-c~,a /2 /nn+l \ in R~ +1, where ~ijuo,~ ~rt T ), r [a T ), AoEC~'~/2(R~ +1) and A--const. Suppose the coefficients of the system (2.28) do not depend on t in the layer R~ +1, O<6<T. Let the matrix-valued functions Ajk(x, 0), J4i(x, 0), ~40(x, 0), denoted by Ajk(X), flU(x), Jlo(X), respectively, satisty the conditions (i), (ii) of Theorem 2.4. Suppose further that Iv(x,0)l<l . Then ]v(x,t)[<_l in R~ +1. By M we denote the value/C(R ~, T) for the system (2). Since u(x, t)=v(x, t)exp(A2t) is the solution of (2), then

sup{ [v(x, t)[: (x, t) e R~, +1 \ R ~ +1 } _< M exp(-A26).

Thus, the solution of the Cauchy problem for the system (2.28) satisfies the classical maximum modulus principle for sufficiently large values of A. []

2.2. T h e case of comp lex coeff ic ients

We can extend the results of the first subsection to the systems (2), (3) with complex coefficients and solutions u=v+iw, where v and w are m-component vector-valued functions with real components. The results are obtained by applica- tion of the corresponding assertions on the maximum modulus for the real case to systems obtained by the separation of real and imaginary parts (see Subsection 1.2). Thus we can formulate analogous theorems and corollaries as in Subsection 2.1. By C m we denote the complex linear m-dimensional space with the inner product (., "/.

We retain the notations of Subsection 1.2 and use them putting s = l . By analogy with the definition (2.1) of the constant ~(fl , T) let

IlulIc( ) /E'(ft, T ) = s u p Ii u ]I% llm(~r) '

where the supremum is taken over all vector-valued functions u : v + i w in the class C(2,1)(QT)nC(QT) that satisfy the system (2) with complex coefficients. Here v and w are m-component vector-valued functions with real components.

T h e o r e m 2.1'. The classical maximum modulus principle holds for solutions of the system with complex coefficients

O2u + Aj(t) Ou 0t =o

j , k= l j = l


in R~ +1 if and only if the equalities

Ajk(t)=ajk(t)Im, Aj( t)=aj( t)I ,~, l <_j,k<n,

are valid, where ( (ajk) ) is a real positive-definite (n x n)-matrix-function and aj are real scalar functions.

T h e o r e m 2 .2 / . Let the classical maximum modulus principle be valid for the system (2) with eomplex eoeffcients in Qr(R +I). Then:

(i) for all x E ~ (xER ~) the equalities

Ajk(X,O)=ajk(X)Im, l < j , k < n ,

hold, where ( (ajk ) ) is a real positive-definite (n • n)-matrix-valued function; (ii) for all xC~ (xER n) and all ~j ,~cC m, j = l , ...,n, with R e ( ~ j , ~ ) = 0 , the

inequality

Re ajk(x)(~y,~k)+E(Aj(x,O)~j,;)+(Ao(x,O)g,; ) >0 "j,k=l j = l

is valid.

T h e o r e m 2 .4 I. The classical maximum modulus principle is valid for solutions - .Rn+l~ of the system (3) with complex coefficients in QT( T j if and only if

(i) for all xE~ (x~R n) the equalities

Ajk(x)=ajk(x)Im, l<_ j , k<n ,

hold, where ( (ajk ) ) is a real positive-definite (n • n)-matrix-valued function; (ii) for all xCa (xER '~) and for all ~j, ;EC "~, j --1, . . . ,n , with R e ( ~ j , ; } = 0 ,

the inequality

Re ajk(x)(~j ,~k)+ (Aj(x)~j,~}+(,Ao(X)q,~} ~_0 ~j,k~l j = l

is valid.

C o r o l l a r y 2 .1 / . The classical maximum modulus principle is valid for the system

0 u Ou ou Aj (x) + J = o Ot ~ .= j,k=l

with complex coefficients in QT(R~ +~) if and only if for all xE~ (xCR n) the equalities

Ajk(x)=aje(X)Im, A~(x)=aj(x)Im, l< j , k<_n ,

hold, where ((ajk) ) is a real positive-definite (nx n)-matrix-valued function and aj are real scalar functions.


C o r o l l a r y 2.2q The classical maximum modulus principle is valid for solutions of the system (3) with complex coefficients in QT(R~ +1) if and only if the condition (i) of Theorem 2.41 is satisfied and

(ii') for all xE~2 (xER ~) and for any gEC m, I~l--1 the inequality

f i bit(x) Re(~4i(x)r ~) Re(~4t(x)q, ; ) - f i bij(x)(,4*(x)q, ~4~(x)r i , j = l i , j = l

+4 ae(A0(x); , ;) > 0

holds. Here ((btk(x))) is the (n• inverse of ((ajk(x))) and ~4~(x) is the adjoint matrix of ~4j (x).

We remark that the second sum is real by the symmetry of the matrix ((bit (x)). In particular, the next assertion follows from Corollary 2.2 t for the scalar par-

abolic equation with complex coefficients

(2.29) Ou 0 u atk(x) + aS(x) +a0(x)u=0 Ot

j , k = l 3 ~ j = l

Coro l l a ry 2.3q The classical maximum modulus principle is valid for (2.29) in QT(R~ +1) /f and only if:

(i) the (n • n)-matrix-valued function ( (ajk (x))) is real and positive-definite (ii) for all xE~ (xER ~) the inequality

4 Re ao (x) >_ f i bjk (x) Im aj (x) Im ak (x) j , k = l

holds.

R e f e r e n c e s

1. AMANN, H., Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. Math. Anal. Appl. 65 (1978), 432-467.

2. BEBERNES~ J. and SCHMITT, K., Invariant sets and the Hukuhara-Kneser property for systems of parabolic partial differential equations, Rocky Mountain J. Math. 7 (1977), 557-567.

3. CHUEH, K. N., CONLEY, C. C. and SMOLLER, J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), 373-391.

4. COSNER, C. and SCHAEFER, P. W., On the development of functionals which satisfy a maximum principle, Appl. Anal. 26 (1987), 45-60.


5. EIDEL'MAN, S. D., Parabolic Systems, North-Holland, Amsterdam, 1969. 6. MAZ'YA, V. G. and KRESIN, G. I., On the maximum principle for strongly elliptic

and parabolic second order systems with constant coefficients, Mat. Sb. 125 (1984), 458-480 (Russian). English transl.: Math. USSR-Sb. 53 (1986), 457- 479.

7. MIRANDA, C., Sul teorema del massimo modulo per una classe di sistemi ellittici di equazioni del secondo ordine e per le equazioni a coefficienti complessi, Istit. Lombardo Accad. Sci. Lett. Rend. A 104 (1970), 736-745.

8. PROTTER, M. H. and WEINBERGER, H. F., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967.

9. REDHEFFER, a . and WALTER, W., Invariant sets for systems of partial differential equations. I Parabolic equations, Arch. Rational Mech. Anal. 67 (1978), 41- 52.

10. STYS, T., On the unique solvability of the first Fourier problem for a parabolic system of linear differential equations of second order, Comment. Math. Prace Mat. 9 (1965), 283-289 (Russian).

11. WALTER, W., Differential and Integral Inequalities, Springer-Verlag, New York, 1970. 12. WEINBERGER, H. F., Invariant sets for weakly coupled parabolic and elliptic systems,

Rend. Mat. 8 (1975), 295-310.

Received September 4, 1992 Gershon I. Kresin The Research Institute The College of Judea and Samaria Kedumim-Ariel, D.N. Efraim 44820 Israel

Vladimir G. Maz'ya Department of Mathematics Link5ping University S-581 83 LinkSping Sweden

criteria for validity of the maximum modulus principle for

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